5.17 Phase portraitsIB Maths: Applications and Interpretation HL: Mind map

Exam questions on 5.17 Phase portraits

1. The populations $x$ and $y$ (in hundreds) of two species of algae in a pond, $t$ weeks after they are introduced, satisfy $\frac{dx}{dt}=4x+y$ and $\frac{dy}{dt}=2x+3y$. A GDC may be used.
   Find an eigenvector of the matrix corresponding to the eigenvalue $2$.2 marks
2. The deviations $x$ and $y$ (in thousands) of the populations of two interacting species from their equilibrium values satisfy $\frac{dx}{dt}=x+2y$ and $\frac{dy}{dt}=3x$. A GDC may be used.
   Find the equations of the two straight-line trajectories that pass through the origin.2 marks
3. The deviations $x$ and $y$ (in hundreds) of two interacting populations from their equilibrium values, $t$ years after a survey begins, satisfy $\frac{dx}{dt}=-x-4y$ and $\frac{dy}{dt}=x-y$.
   Find the eigenvalues of the matrix $\begin{pmatrix} -1 & -4 \\ 1 & -1 \end{pmatrix}$ that represents the system.3 marks